e_120230
plain-language theorem explainer
Pointwise identity: the folded Regge numerator m2Num at Fin-4 indices (1,2,0,2,3,0) equals eight times the explicit integer table explicitZ at the same indices. Gravity analysts cite it as one of 256 kernel cells in the 4D midpoint M2 TT identity. The proof is a single kernel decide on two concrete integers.
Claim. For indices $(a,b,c,d,i,j)=(1,2,0,2,3,0)$ in $(\mathrm{Fin}\,4)^6$, the folded coupling numerator $m_2^{\mathrm{num}}(a,b,c,d,i,j)$ equals $8$ times the explicit integer lookup $Z(a,b,c,d,i,j)$.
background
This module is chunk 6 of a 256-cell kernel certification that the 4D Regge midpoint $M_2$ numerator equals eight times a closed-form integer table. Indices run over $\mathrm{Fin},4$ (four spacetime directions).
The numerator $m_2^{\mathrm{num}}$ is defined by folding a fixed coupling list: start at $0$ and add each contribution $\mathrm{contrib}(t,a,b,c,d,i,j)$. The table $Z$ is an explicit pattern-match on the six indices, returning small integers such as $4$, $-2$, and so on for the nonzero support.
The local claim is only the single sextuple $(1,2,0,2,3,0)$. Sibling theorems cover the other cells in the same chunk.
proof idea
One-line kernel proof: decide. Both sides reduce to concrete integers (the fold of m2Num at fixed indices versus 8 * explicitZ at those indices), so the decidable equality on Int closes the goal with no lemmas or case splits.
why it matters
Feeds the assembly theorem m2Num_eq_eight_explicitZ, which states the identity for every sextuple in $(\mathrm{Fin},4)^6$ by exhausting indices with fin_cases. That global equality is the certified algebraic core of the 4D Regge exact-midpoint $M_2$ TT identity in the Gravity analysis stack. Without the pointwise cells, the assembly cannot discharge all $4^6$ cases. It is pure discrete tensor algebra on the lattice, not a continuum GR derivation, and sits downstream of the kernel certificate definitions rather than the T0–T8 forcing chain.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.